Question

Choose the correct answer in the given question.
$\int e^{x}secx(1+tanx)dx$ equals
$\left(a\right)e^{x}cosx+C\left(b\right)e^{x}secx+C\left(c\right)e^{x}sinx+C\left(d\right)e^{x}tanx+C$

Answer 1

Let $I=\int e^{x}secx(1+tanx)dx$
$\Rightarrow I=\int e^{x}secxdx+\int e^{x}secxtanxdx..\left(i\right)$
Now, $\int e^{x}secxdx=secx\int e^{x}dx-\int [\frac{d}{dx}secx\int e^{x}dx]dx$
$=e^{x}secx-\int secxtanx{e}^{x}dx......\left(ii\right)$
On putting the value from Eq. (ii) in Eq. (i), we get
$I=e^x$ sec x $-\int e^x$ sec x tan x dx +$\int$ sec x tan x $e^{x}dx+C$
$\Rightarrow I=e^x$ sec x+C. Hence, correct option is (a).